New Routing Techniques and their Applications

Abstract

In this paper we present two new routing techniques that allow us to obtain the following new routing schemes: A routing scheme for n-nodes, m-edges unweighted graphs that uses O(1/e n2/3) space at each vertex and O(1/e)-bit headers, to route a message between any pair of vertices u,v ∈ V on a (2 + e,1)-stretch path, i.e., a path of length at most (2 + e)• d+1, where d is the distance between u and v. This should be compared to the (2,1)-stretch and O(n5/3) space distance oracle of patrascu and Roditty [FOCS'10 and SIAM J. Comput. 2014] and to the (2,1)-stretch routing scheme of Abraham and Gavoille [DISC'11] that uses O(n3/4) space at each vertex. It follows from patrascu, Thorup and Roditty [FOCS'12] that a 2-stretch distance oracle with O(m2/3) space at each vertex is optimal, assuming a hardness conjecture on set intersection holds. A routing scheme for n-nodes weighted graphs with normalized diameter D, that uses O(1/e n1/3log D) space at each vertex and O(1/e log D)-bit headers, to route a message between any pair of vertices on a (5+e)-stretch path. This should be compared to the 5-stretch and O(n4/3) space distance oracle of Thorup and Zwick [STOC'01 and J. ACM. 2005] and to the 7-stretch routing scheme of Thorup and Zwick [SPAA'01] that uses O(n1/3) space at each vertex. Since a 5-stretch routing scheme must use tables of Ω(n1/3) space our result is almost tight. For an integer l>1, a routing scheme for n-nodes unweighted graphs that uses O(l 1/e nl/(2 l pm 1)) space at each vertex and O(1/e)-bit headers, to route a message between any pair of vertices on a (3 pm 2 / l + e,2)-stretch path. This should be compared to the distance oracles of patrascu, Thorup and Roditty [FOCS'12] for weighted graphs with a stretch of (3 pm 2/l) and O(l m1+l/(2 l pm 1)) total space. A routing scheme for n-nodes weighted graphs, that for any integer k>2, uses O(1/e n1/klog D) space at each vertex and O(1/e log D)-bit headers, to route a message between any pair of vertices on a (4k-7+e)-stretch path. This improves the (4k-5)-stretch routing scheme of Thorup and Zwick [SPAA'01] and can also be used in the recent ((4 - α)k - β)-stretch routing scheme of Chechik [PODC'13] to obtain slightly better values for α and β.